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Theorem isclat 15613
Description: The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
isclat.b  |-  B  =  ( Base `  K
)
isclat.u  |-  U  =  ( lub `  K
)
isclat.g  |-  G  =  ( glb `  K
)
Assertion
Ref Expression
isclat  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )

Proof of Theorem isclat
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . . . 6  |-  ( l  =  K  ->  ( lub `  l )  =  ( lub `  K
) )
2 isclat.u . . . . . 6  |-  U  =  ( lub `  K
)
31, 2syl6eqr 2526 . . . . 5  |-  ( l  =  K  ->  ( lub `  l )  =  U )
43dmeqd 5211 . . . 4  |-  ( l  =  K  ->  dom  ( lub `  l )  =  dom  U )
5 fveq2 5872 . . . . . 6  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
6 isclat.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2526 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  B )
87pweqd 4021 . . . 4  |-  ( l  =  K  ->  ~P ( Base `  l )  =  ~P B )
94, 8eqeq12d 2489 . . 3  |-  ( l  =  K  ->  ( dom  ( lub `  l
)  =  ~P ( Base `  l )  <->  dom  U  =  ~P B ) )
10 fveq2 5872 . . . . . 6  |-  ( l  =  K  ->  ( glb `  l )  =  ( glb `  K
) )
11 isclat.g . . . . . 6  |-  G  =  ( glb `  K
)
1210, 11syl6eqr 2526 . . . . 5  |-  ( l  =  K  ->  ( glb `  l )  =  G )
1312dmeqd 5211 . . . 4  |-  ( l  =  K  ->  dom  ( glb `  l )  =  dom  G )
1413, 8eqeq12d 2489 . . 3  |-  ( l  =  K  ->  ( dom  ( glb `  l
)  =  ~P ( Base `  l )  <->  dom  G  =  ~P B ) )
159, 14anbi12d 710 . 2  |-  ( l  =  K  ->  (
( dom  ( lub `  l )  =  ~P ( Base `  l )  /\  dom  ( glb `  l
)  =  ~P ( Base `  l ) )  <-> 
( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )
16 df-clat 15612 . 2  |-  CLat  =  { l  e.  Poset  |  ( dom  ( lub `  l )  =  ~P ( Base `  l )  /\  dom  ( glb `  l
)  =  ~P ( Base `  l ) ) }
1715, 16elrab2 3268 1  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   ~Pcpw 4016   dom cdm 5005   ` cfv 5594   Basecbs 14507   Posetcpo 15444   lubclub 15446   glbcglb 15447   CLatccla 15611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-dm 5015  df-iota 5557  df-fv 5602  df-clat 15612
This theorem is referenced by:  clatpos  15614  clatlem  15615  clatlubcl2  15617  clatglbcl2  15619  clatl  15620  oduclatb  15648  xrsclat  27492
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